Intro Lin Alg Elimination problem

[brushman]

Homework Statement
Look for a matrix that has row sums 4 and 8, and column sums 2 and x:

matrix =
[a, b] -------I don't know how to do latex
[c, d]

a + b = 4; a + c = 2;
c + d = 8; b + d = s;

The four equations are solvable only if s = ____. Then find two different matrices that have the correct row and column sums. Write down the 4 by 4 system Ax = b with x = (a, b, c, d) and make A triangular by elimination.

Attempt at a solution:

I created a matrix from the system of equations given. I started with:

[1 1 0 0 4]
[0 0 1 1 8]
[1 0 1 0 2]
[0 1 0 1 s]

-->[1 1 0 0 4]
[0 -1 1 0 -2]
[0 0 1 1 8]
[0 0 1 1 (s-2)]

-->

[1 1 0 0 4]
[0 -1 1 0 -2]
[0 0 1 1 8]
[0 0 0 0 (s-10)]

So I got s = 10.
I can't solve for a, b, c, or d (or can I?)

What am I not understanding and how do I do this problem? Thanks.

edit: Thanks LCKurtz, I got it now.

 

[LCKurtz]

Homework Statement
Look for a matrix that has row sums 4 and 8, and column sums 2 and x:

matrix =
[a, b] -------I don't know how to do latex
[c, d]

a + b = 4; a + c = 2;
c + d = 8; b + d = s;

The four equations are solvable only if s = ____. Then find two different matrices that have the correct row and column sums. Write down the 4 by 4 system Ax = b with x = (a, b, c, d) and make A triangular by elimination.

Attempt at a solution:

I created a matrix from the system of equations given. I started with:

[1 1 0 0 4]
[0 0 1 1 8]
[1 0 1 0 2]
[0 1 0 1 s]

-->


[1 1 0 0 4]
[0 -1 1 0 -2]
[0 0 1 1 8]
[0 0 1 1 (s-2)]

-->

[1 1 0 0 4]
[0 -1 1 0 -2]
[0 0 1 1 8]
[0 0 0 0 (s-10)]

So I got s = 10.
I can't solve for a, b, c, or d (or can I?)

What am I not understanding and how do I do this problem? Thanks.

Sure you can solve for them. Once you assign s =10, the last equation becomes dependent on the others. You have more variables than equations so you would expect infinitely many solutions. Let d = d (anything) and solve for the others in terms of d by working back up the system. For example your third equation says c + d = 8 so c = 8 - d. So b = ... etc. Make sure you don't have an arithmetic mistake in your reduction (I think you do.) Once you have a, b, and c in terms of d, check that they work in your system and you can continue with the problem.